Example 12 illustrates the ease of model fitting when

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Example 1.2 illustrates the ease of model fitting when binary-valued covariates are known; the notion of permutation similarity plays a similar role in the case of k -ary covariates. 1.3.2. Approximate inference The careful reader will have noted that in the case of known categorical covariates, examples such as those above can be expressed as contingency tables – a notion we revisit in Section 1.4 – and hence may admit exact inference procedures. However, if covariates are latent, then an appeal to maximum-likelihood estimation induces a combinatorial optimization prob- lem; in general, no fast algorithm is known for likelihood maximization over the set of covariates and Bernoulli parameters under the general k -group stochastic block model. The principal difficulty arises in maximizing the n -dimensional k -ary covariate vector c over an exponentially large model space; estimating the Copyright © 2014. Imperial College Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/16/2016 3:37 AM via CGC-GROUP OF COLLEGES (GHARUAN) AN: 779681 ; Heard, Nicholas, Adams, Niall M..; Data Analysis for Network Cyber-security Account: ns224671
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10 B. P. Olding and P. J. Wolfe Fig. 1.3. Representations A and Π A Π of data drawn from the stochastic block model of Example 1.3, corresponding to isomorphic graphs (black boxes denote links). Though p 01 = 0, only a small subset of permutation similarity transformations Π ( · ) Π will reveal the disconnected nature of this network. ( k +1 2 ) associated Bernoulli parameters then proceeds in exact analogy to Example 1.2 above. The following example illustrates the complexity of this inference task. Example 1.3 (Permutation and Maximization). Consider a 100- node network generated according to the stochastic block model of Defini- tion 1.2 , with each group of size 50 and p 00 = p 11 = 1 / 2 , p 01 = 0 . Figure 1.3 shows two permutation-similar adjacency matrices, A and Π A Π , that cor- respond to isomorphic graphs representing this network ; inferring the vector c of binary categorical covariates from data A in Figure 1.3a is equivalent to finding a permutation similarity transformation Π A Π that reveals the distinct division apparent in Figure 1.3b. Given the combinatorial nature of this problem in general, it is clear that fitting models to real-world network data can quickly necessitate approximate inference. To this end, Example 1.3 motivates an important means of exploiting algebraic properties of network adjacency structure: the notion of a graph spectrum . Eigenvalues associated with graphs reveal several of their key properties (Chung, 1997) at a computational cost that scales as the cube of the number of nodes, offering an appealing alternative in cases where exact solutions are of exponential complexity.
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